Mean free path and number of collisions between molecules

3 Mean-free path and number of collisions between molecules 

To understand heat conduction, diffusion, viscosity and chemical kinetics the mechanistic view of molecule motion is of fundamental importance. The fundamental quantity is the mean-free path, i. e. the distance of a molecule between two collisions with any other molecule. The number of collisions between a molecule and a wall was shown in Chapter 4.1.1.2 to be z = CNQvdtl6. Similarly, we can calculate the number of collisions between molecules from a geometric view. We denote that all molecules have the mean speed v and their mean relative speed with respect to the colliding molecule is g. When two molecules collide, the distance between their centers is d in the case of identical molecules, d corresponds to the effective diameter of the molecule. Hence, this molecule will collide in the time dt with any molecule centre that lies in a cylinder of a diameter 2d with the area Jid and length gdt (it follows that the volume is Jtd gdt). The area where d is the molecule (particle) diameter is also called collisional cross section a. This is a measure of the area (centered on the centre of the mass of one of the particles) through which the particles cannot pass each other without colliding. Hence, the number of collisions is z = c n gdt. A more correct derivation, taking into account the motion of all other molecules with a Maxwell distribution (see below), leads to the same expression for z but with a factor of V2. We have to consider the relative speed, which is the vector difference between the velocities of two objects A and B (here for A relative to B)  

Since vXvb must average zero, the relative directions being random, the average square of the relative velocity is twice the average square of the velocity of A + B and, therefore, the average root mean square velocity is increased by a factor V2 (remember that v), and the collision rate is increased by this factor 

To determine the distance travelled between collisions, the mean-free path /, we must divide the mean molecule velocity by the collision frequency or, in other words, the mean travelling distance vdt by the number of collisions. Furthermore, we must find an expression for the collision frequency (v = z dt) and the mean-free time (r = 1/ v)  

That fraction of gas molecules is of interest when velocity is within v and v + dv. This fraction f v)dv is time-independent from the exchange with other molecules. The Boltzmann distribution states that at higher energies (= kT) the probability to meet molecules is exponentially less  

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